Consider the sequence of positive integers defined by $s_1,s_2,s_3, \dotsc $ of positive integers defined by $s_1=2$, and for each positive integer $n$, $s_{n+1}$ is equal to $s_n$ plus the product of prime factors of $s_n$. The first terms of the sequence are $2,4,6,12,18,24$. Prove that the product of the $2019$ smallest primes is a term of the sequence.